Some Formulas and Tables

Abstract

This chapter contains a list of formulae from the calculus on manifolds, and tables concerning Lie groups, Lie algebras and symmetric spaces, as a list of the Poincaré polynomials of compact simple Lie groups, an overview of real forms of classical complex simple Lie algebras and their corresponding simple Lie groups, a table of irreducible Riemannian symmetric spaces of type I and III, a table of Riemannian symmetric spaces of classical type with noncompact isotropy group, etc.)

One can find the formula for Christoffel symbols, the curvature tensor, Bianchi identities, Ricci tensor, the basic differential operators, the expression for conformal changes of Riemannian metrics, Cartan structure equations for pseudo-Riemannian manifolds, etc.

Many of these formulas are used throughout the book; others are not, but they have been included since such a collection should be useful as an aide-mémoire, even for teachers and researchers.